Summary Our project address fundamental problems of structures and mechanisms of protein molecules and their interaction networks. At the protein level, we will continue our NIGMS supported studies and focus on A) ?-Barrel Membrane Proteins (?MPs). We will develop models and computational tools for predicting their structure, understanding their mechanism, and formulating design of novel ?MPs. At the network level, we will explore a new research direction. We will B) develop models and exact algorithms by solving the discrete chemical master equation (dCME) to compute exact probability landscape and discrete probability flux of networks for studying stochastic control of cellular behavior. Furthermore, we will examine how phenotype switching arise in networks, starting from commonly occurring network motifs and comprehensively characterize the universe of their multistabilities. In Project A), we will focus on ?MPs found in the outer membrane of gram-negative bacteria, eukaryotic mitochondria, and in exotoxins. ?MPs are involved in fundamental processes such as transport, translocation, energy metabolism, and apoptosis induction. They are also important therapeutic targets against infectious diseases. In addition, there are significant engineering interests in developing ?MPs as bionanopores for single molecule detection and other biotech applications. Despite recent progress, our knowledge of ?MPs is limited: only a few dozens of structures of non-homologous ?MPs are known. Importantly, we lack general understanding of the organizing principles of ?MPs and their functioning mechanism. We propose to develop models and algorithms for A1) predicting structures of ?MPs, A2) deciphering the mechanism of gating in OmpG, and A3) designing novel ?MPs with desired geometry and stability towards broad biotech applications. Our approach will be based on the reduced state model we developed, the MHIP empirical potential function we obtained through extensive combinatorial analysis, the (m)DiSGro loop structure prediction and sampling algorithms, with significant new development. In Project B), we will focus on the fundamental problem of constructing exact stochastic probability landscape of networks of interacting molecules. Many important biological reactions involve only a small copy number of molecules. Stochasticity arising from such low copy events as well as rare events are important for fundamental processes such as embryonic development, stem cell differentiation and nongenetic heterogeneity. While the discrete chemical master equation (dCME) provides a generate framework for understanding stochasticity in mesocopic systems, many foundational problems remain. Despite significant progress, the exact time- evolving probability landscapes for many networks of interests are computationally inaccessible, except for a few simple toy problems (e.g. those with <4 nodes). One has to rely on Gillespie simulation or approximation of Langevin/Fokker-Planck formulations, with errors largely unexamined. We propose to develop B1) theoretical model and tools for computing probability fluxes on discrete state space at arbitrary microstate and for arbitrary reactions, so passageways, transient states, and critical paths important for characterizing phenotypical switches can be identified. We will also carried out computational analysis to decipher B2) common mechanisms of stochastic switching, as well as comprehensive mapping of phase diagrams of emerging multistabilities in the most widely encountered common biological motifs. Our approach will be based on our recent algorithm and theoretical development of multi-finite buffer network structure analysis, the corner simplex optimal state enumeration algorithm, and the ACME method for exact computation of time-evolving probability landscape, as well as error-bound analysis based our quotient matrix and the technique of stochastic ordering.